j 
198 
indeterminate analvsis. Euler and M. 
Lagrange have found them. M. Legen- 
dre added to them several important 
propositions; and in his Essay on the 
Theory of Numbers, resumed the sub- 
ject from its origin, and undertook pro- 
found researches to obtain the demon- 
stration, then unknown, of the general. 
theorem of Fermat. M. Gauss has 
treated this whole theory in a manner 
entirely new, in a singularly remarkable 
work, of which it is not in our power to 
convey an idea, because the whele of it 
is new, even the language and notation. 
To this kind of analysis may be referred 
the theory of continual fractions, and 
that of the transformation of equations, 
so successfully treated of by M. Lagr anve. 
The differential and integral calculus 
dceupied geometricians for a hundred 
years; and l’'Hopital’s Infiniment Petits, 
and Bougainville’s Integral Calculus, 
were the only works which formed a 
system. Euler has since given more 
complete treatises, which he enriched by 
Ins discoveries; the ‘rapid progress of 
analysis rendered them insufficient (39). 
M. Lacroix, who devoted himself to 
teaching, collected in one large treatise 
all the scattered methods, by connecting, - 
and developing them ; by adding his own 
ideas to theirs he has associated him- 
self to the glory of the great geome- 
tricians whose discoveries he propagated. 
M. Bossut, so well known by his 
tracts on all the parts of the mathematics, 
and by his Hydrodynamics, of which he 
has lately given a new edition, with 
ditions has published a History of 
M athematics, which renders the conti- 
nuation, promised by the author, very 
desirable. ML. de Montucla rendered 
himself celebrated by a more detailed 
history, which he could not resume till 
towards the end of his life; le was not 
able even to complete it, and Lalande 
has filled up the chasms. 
More attention was paid to the ex- 
tension of the calculation of infinites 
than to the explanation of the meta- 
physical principles of it ; the miraculous 
effects, the incontrovertible results, were 
seen, but the mind could not accustom 
itself to the fundamental suppositions. 
M,. Lagrange, in a celebrated memoir, 
laid down one of those comprehensive 
- ideas, which belong only to men of genius 
of the first order; he pointed out the 
means of reducing all the processes of 
the infinitesimal calculations 
merely algebraical, by carefully excluding 
aliidea of infinity. Many geometricians, 
Progress of the Sciences. - 
to one 
[sept. 1, 
strabk with this flash of light, sought for 
the illustrations, which none wanted supply- 
so well as the inventor. M. Lagrange, 
having undertaken the funetions of 
teacher at the polytechnic school, 
created there, in the presence of his 
auditors, all the parts of which he has 
since composed his Treatise on Atialy- 
tical Functions, a classic work, which it 
would be superfluous at present to re- 
commend ; it is sufficient to have menti- 
oned it. ‘The same principles served to ex= 
plain the metaphysical part of the calcu- “ 
Jation of the variations, which from the- 
begiuning of his career ranked him with in= 
ventive geometricians, and the use of 
which has tately been extended by Mr. 
Poisson, who shews an elegantand simple 
manner of obtaining the imdeterminate 
equations resulting from this method. 
The calculation of partial differences, 
respecting which Euler and D’Alembert 
did not agree, and the utility of which is 
equal to the innumerable difficulties which 
it presents, has given rise to the researches 
of all the eminent geometricians known: 
MM. Laplace and Condorcet thought 
of considering the equations containing 
at the same time differential co-efficients 
and differences, which M. Lacroix has 
distinguished by the name of equations 
with mixt differences. M. Biot has given 
some yeneral principles for the solution 
of this species of integrals. MM. 
Poisson and Paoli have extended still 
farther this theory, which more than 
any other is impossible to be translated 
into ordinary language. 
All the laws of mechanics have been 
reduced to general principles, of which 
we shall mention that of virtual veloci- 
ties, the only basis of M. Lagrange’s 
analytical mechanics, which he has been 
able, with the assistance of the doctrine 
of the variations, to apply to every case 
of equilibrium and motion. 
M. Lagrange had first assuined this 
principle; he has since given a demon- 
stration of it; another, by Laplace, is to 
be found in the Celestial Mechanics ; and 
MM. Poinsot and Ampere have diss 
covered others. ‘There existed one’ more 
ancient in the treatise on equilibrium 
and motion, by M. Carnot. MM. 
‘Prony and Poisson, in their lectures at 
the polytechnic school, have had more 
than one Dp POT Len Ey er occupying them- 
selves with analogous researches. 
M. Laplace reduced to the same prin- 
ciple his numerous researches on the 
system of the world. He resumed the 
doctrine of mechanics from its first prin- 
ciples, 
is 
